Convex programming techniques were used by Witting and Krafft in [4] in order to reduce a testing problem for composite hypotheses to one for simple hypotheses. This is realized in terms of least favourable pairs of distributions, which represent the solution of the dual of a suitable program. Without further assumptions on the hypotheses, however, the results, derived that way (cf. Baumann [1], 0sterreicher [6] and Kusolitsch and Osterreicher [5]), are of less practical impact. This is due to the fact that in this case the least favourable pairs depend on the level of the testing problem. Conditions avoiding this, were given by Huber and Strassen in [-3]. These conditions make use of 2-alternating capacities in the sense of Choquet. The present paper offers a rather general principle of constructing the least favo urable distribution in the case, when one of the two hypotheses is simple. This method works also for the Iocal variation model and the Prohorov neighbourhood model in the case of monotone likelyhood ratio. For "simple" cases -subsuming the gross error model and the total variation model, for which the solution was given by Huber in [2] -a least favourable pair is obtained by using the mentioned technique of construction two times successively.
At the core of this paper is a simple geometric object, namely the risk set of a statistical testing problem on the one hand and f -divergences, which were introduced by Csiszár (1963) on the other hand. f -divergences are measures for the hardness of a testing problem depending on a convex real valued function f on the interval [0, ∞). The choice of this parameter f can be adjusted so as to match the needs for specific applications.
Let P=(p1, ..., pn)be a probability distribution on a setΩ={ω1, ..., ωn}with n elements, n ∈ N\{1}. Then the termS2(P)=1−Íni=1p2i,frequently called the Gini–Simpson index, or, in information theory, quadratic entropy, is used in many different areas of research resp. applications and was, therefore, reinvented several times. In this note we give aconcise history of this index and closely related measures, as well as its generalisation to allvalues of the parameter of the class of entropies of orderα∈(0,∞)\{1}introduced by Havrda and Charvát(1967) and reinvented by Tsallis(1988) for the use of this index in statistical physics, for which the limiting case forα→1isShannon’s entropyS1(P)=−Íni=1pilnpi.We also give a brief note on weighted versions of the Gini–Simpson index.In addition to these central historic features our note also presents contributions on the axiomatics of entropies and on the early history of the application of the concept of entropyin thermodynamics. We also provide an entry on Rényi’s class of entropies, linked with Hill’s diversity numbers
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